CBE2027 Structural Analysis I Chapter 7 – Mohr’s Circle HD in Civil Engineering Page 7 -1 MOHR’S CIRCLE A force is easily resolved to components in another coordinate system by a simple trigonometric calculation. Transformation of stress to another coordinate system is also a purely mathematical process but is more involved than the transformation of force. Stress not only has a magnitude and a direction, but is also associated with an area over it acts. A transformation of coordinates for stress will change the area of the orientation plane. This must be taken into account in stress transformation. Sometimes we are interested in the state of stress at a particular inclined plane, e.g. finding the normal and shearing stresses in an inclined glued splice. More often, we are interested in the maximum intensities of the normal and shear stresses at a point in the member. For this more general case, we also need to determine the orientation of the planes where the maximum normal and shear stresses occur. These are problems to be addressed in stress transformation.
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CBE2027 Structural Analysis I Chapter 7 – Mohr’s Circle
HD in Civil Engineering Page 7 -1
MOHR’S CIRCLE
A force is easily resolved to components in another coordinate system by a
simple trigonometric calculation. Transformation of stress to another
coordinate system is also a purely mathematical process but is more
involved than the transformation of force.
Stress not only has a magnitude and a direction, but is also associated with
an area over it acts. A transformation of coordinates for stress will change
the area of the orientation plane. This must be taken into account in stress
transformation.
Sometimes we are interested in the state of stress at a particular inclined
plane, e.g. finding the normal and shearing stresses in an inclined glued
splice. More often, we are interested in the maximum intensities of the
normal and shear stresses at a point in the member. For this more general
case, we also need to determine the orientation of the planes where the
maximum normal and shear stresses occur. These are problems to be
addressed in stress transformation.
CBE2027 Structural Analysis I Chapter 7 – Mohr’s Circle
HD in Civil Engineering Page 7 -2
N = P cos θθθθ V = P sin θθθθ
The normal force N produces positive normal stresses σθ and the shear force
V produces negative shear stress τθ. These stresses can be evaluated by
dividing the forces by the area over which they act. The area A1 of the
inclined section is A/cos θ, in which A is the cross-sectional area.
σθ= N/A1 = (P/A)* cos2 θ = σx cos2 θ
τθ= -V/A1 = -(P/A) sin θ cos θ = -σx sin θ cos θ
in which σx = P/A is the normal stress on a cross-section.
CBE2027 Structural Analysis I Chapter 7 – Mohr’s Circle
HD in Civil Engineering Page 7 -3
STRESS TRANSORMATION EQUATIONS
An element is subject to the following stresses. We are interested in the
maximum intensities of the normal and shear stresses in the element. For
this case, we need to determine the orientation of the planes where the
maximum normal and shear stresses occur.
Let the thickness of this element ( in the z-direction) be equal to unity.
∑Fn=0
σθds - σx dy cos θ - σy dx sin θ + τx dy sin θ + τy dx cos θ = 0